When are off-diagonal hypergraph Ramsey numbers polynomial?

TL;DR

This paper investigates when off-diagonal hypergraph Ramsey numbers grow polynomially, providing characterizations for certain classes of 3-graphs based on their structural properties.

Contribution

It establishes a criterion linking polynomial growth of off-diagonal Ramsey numbers to containment in iterated blowups of an edge for specific 3-graphs.

Findings

Polynomial growth characterized for tightly connected 3-graphs.

Polynomial growth linked to containment in iterated blowups.

Provides a structural criterion for off-diagonal hypergraph Ramsey numbers.

Abstract

A natural open problem in Ramsey theory is to determine those $3$-graphs $H$ for which the off-diagonal Ramsey number $r(H, K_n^{(3)})$ grows polynomially with $n$. We make substantial progress on this question by showing that if $H$ is tightly connected or has at most two tight components, then $r(H, K_n^{(3)})$ grows polynomially if and only if $H$ is contained in an iterated blowup of an edge.